## The Annals of Probability

### The First Birth Problem for an Age-dependent Branching Process

J. F. C. Kingman

#### Abstract

If $B_n$ denotes the time of the first birth in the $n$th generation of an age-dependent branching process of Crump-Mode type, then under a weak condition there is a constant $\gamma$ such that $B_n/n \rightarrow \gamma$ as $n \rightarrow \infty$, almost surely on the event of ultimate survival. This strengthens a result of Hammersley, who proved convergence in probability for the more special Bellman-Harris process. The proof depends on a class of martingales which arise from a `collective marks' argument.

#### Article information

Source
Ann. Probab., Volume 3, Number 5 (1975), 790-801.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996266

Digital Object Identifier
doi:10.1214/aop/1176996266

Mathematical Reviews number (MathSciNet)
MR400438

Zentralblatt MATH identifier
0325.60079

JSTOR